If I own example.com/net/org, can I simply tell my customers to type "example" into their browser address bar to get to my site? (i.e. no http://, no dot anything)
What Firefox is doing is using the first result from Google's search results as their destination. If you don't rank Number 1 for your company name when they do their query your customers will go elsewhere which may be a competitor or a website dedicated to mocking or trashing your company. This can happen as a normal fluctuation of Google's results or because someone has simply done a better job of outranking you. Since you don't control Google's results I don't think this is a very good idea.